Asymptotics of eigenprojections of correlation matrices with some applications in principal components analysis

Abstract

SUMMARY The asymptotic distribution of an eigenprojection for a sample correlation matrix is obtained. In particular, it is shown that the rank of the asymptotic covariance matrix depends on distributional parameters in a somewhat complicated manner. The results obtained in this paper can be used to determine this rank. Some applications of the asymptotic distribution of these eigenprojections to inferential problems involving principal components subspaces are given. Many statistical applications involve the analysis of a sample covariance matrix or sample correlation matrix. As a result of the complexity of the exact joint distribution of the elements of these matrices, most of these applications utilise the corresponding asymptotic distributions. Since the asymptotic distribution for the sample covariance matrix is somewhat simpler than that of the sample correlation matrix, covariance-matrix based analyses have received more attention in the literature. Unfortunately, in practice, analyses based on correlation matrices are more prevalent than those based on covariance matrices. As a result, in those situations in which correlation-matrix based procedures have not yet been developed, practitioners will often use procedures that have been developed for a corresponding analysis based on a sample covariance matrix. This results in a procedure which, at best, can only be described as a crude approximation. In this paper, we review some of the asymptotic properties of the sample correlation matrix. In particular, we give the asymptotic distribution of eigenprojections computed from the sample correlation matrix. The associated asymptotic covariance matrix of this distribution is obtained and its rank is determined. Throughout this paper, we focus on the sample correlation matrix computed from the usual sample covariance matrix. However, it should be noted that the asymptotic results obtained can be easily generalised to the correlation matrix computed from any asymptotically normal estimate of the covariance matrix. We also consider some applications which utilise eigenprojections of correlation matrices; these are some inference problems in a principal components analysis of a correlation matrix. In some of these situations, straightforward generalisations of the covariance-matrix based test statistics are not possible because of some complications associated with the rank of the asymptotic covariance matrix of the eigenprojections. In these cases,